On weighted norm inequalities for the Carleson and Walsh-Carleson operator

We prove L(w) bounds for the Carleson operator C, its lacunary version Clac, and its analogue for the Walsh series W in terms of the Aq constants [w]Aq for 1 q p. In particular, we show that, exactly as for the Hilbert transform, ‖C‖Lp(w) is bounded linearly by [w]Aq for 1 q < p. We also obtain L(w) bounds in terms of [w]Ap , whose sharpness is related to certain conjectures (for instance, of Konyagin [International Congress of Mathematicians, vol. II (European Mathematical Society, Zürich, 2006) 1393–1403]) on pointwise convergence of Fourier series for functions near L. Our approach works in the general context of maximally modulated Calderón–Zygmund operators.